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Discontinuous phase unwrapping based on the minimization of Zernike gradient polynomial residual

Xi Yang, Haixiang Hu, Donglin Xue, and Xin Zhang

Open Access Open Access

Abstract

In many certain optical metrology cases, the pupil is usually divided into multiple connected domains by secondary mirror spiders, thus producing segment piston errors and leaving a false phase unwrapping result. In this paper, a method based on minimization of Zernike gradient polynomial residual (MZGR) is proposed to estimate segment piston errors and correct erroneous phase unwrapping results. Simulations and experiments demonstrated that this method can obtain the segment piston errors precisely under complex aberration forms and varied obscurations, indicating reliable practicality. Comparison to the 4D commercial solution, the RMS (root-mean-square) of the residual decreased from 0.154 λ to 0.020 λ.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Interferometry is a broadly established method for wavefront testing of optical mirrors and systems [1–3]. The test results supplied by the interferometer are the fringe patterns, like Fig. 1(a), from which the phase can be generated by the arctangent function. The phase is wrapped from -π to π [4], as shown in Fig. 1(b). Therefore, phase unwrapping is an essential stage in wavefront reconstruction process. In principle, phase unwrapping is to add an appropriate multiple of 2π to the wrapped phase [4]. Phase unwrapping has been widely researched, and several methods have been proposed, including path-following methods [5–8], the least square methods [9–12], transport of intensity equation [13–16], etc.

 figure: Fig. 1.

Fig. 1. (a) fringe pattern; (b) wrapped phase; (c) unwrapped phase.

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But in practice, noise, high dynamic or discontinuity may cause false results in phase unwrapping. In this article, we focus on the phase discontinuity. Common unwrapping methods are based on the assumption that the phase is continuous as a whole [17], like Fig. 1. But in some cases, the continuity can be broken, such as cut by the supporting structure of the secondary mirror, or so-called spider. The discontinuity will cause segment piston errors among segment phases. Figure 2(a) shows a wavefront test result with segment pistons of 4D PhaseCam 6010 with the software 4Sight version 1.4 (build on 2021). Figure 2(b) shows another wavefront test result of ZYGO Verfire HDX interferometer with the software Mx version 8.0 (build on 2021). Both of them indicate neither software solve this problem thoroughly. Some researchers have explored this problem. Bikkannavar (2008) [18] proposed a method to automatically detect wrapped regions in the phase estimate, then replace those regions using valid information from the surrounding pixels to produce a continuous result. Zecchino (2009) [19] bridged phase data in order to accurately measure the mirror as a single, continuous surface based on the slope data. Zong (2021) [17] attempted to bridge the phase basing on the spatial domain utilizing numerical carrier frequency and fringe extrapolation.

 figure: Fig. 2.

Fig. 2. Wavefront test results with segment pistons of commercial interferometers(removed the first nine Zernike polynomials) (a) 4D PhaseCam 6010; (b) ZYGO Verfire HDX.

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In a word, all of these methods are intended to bridge the phase and reconstruct continuity of the data. In the methods of Bikkannavar and Zong, they need to extrapolate the fringe, which means there must be some residue, and the residual may be larger when the fringe is complex and noisy. Moreover, when the phase has aberration, the method of Zecchino may need to adjust manually to get the results right, such as removing the aberration firstly [19], which means that improper operation may result in a wrong result.

This paper proposes a method based on minimization of Zernike gradient polynomial residual (MZGR) to reconstruct the phase with segment pistons. The segment piston is a constant, and we can calculate the gradient of the wrong wavefront results to avoid the segment pistons and use Zernike gradient polynomial to fit the wavefront gradient, so that we can recover the correct unwrapped wavefront and segment piston errors. The result will have less residuals, because there is no process to extrapolate the fringe in the blank areas, which is completely different from the previous studies. And because the number of Zernike gradient polynomial terms can be adjusted autonomously in the calculation process, manual adjustments are not required, even for the phase with complex aberrations. This paper is organized as follows. In Section 2, we illustrate the definition of the MZGR method. In Section 3, we present the performance of this method with complex aberration and varied obscurations. In Section 4, we conduct an experiment to verify the validity and practicality of this method. Finally, we conclude in Section 5.

2. Method based on the minimization of Zernike gradient polynomial residual

2.1 Overview

Spiders or others obscure the mirrors, which causes the phase in the obscured part to be missing. The missing phase leads to phase unwrapping errors and causes some of the phase segments to have independent piston errors which we called segment piston. Due to the error appearing in the phase unwrapping process, which is the process of adding an integer multiple of 2π, segment piston must be an integer multiple of 2π and a constant for the points in the same segment. Figure 3(a) illustrates a fringe pattern divided into multiple segments by spiders. In Fig. 3(b), after calculating with the conventional phase unwrapping method, the phase of the upper right corner segment is obviously higher than the others because of the segment piston error shown in Fig. 3(c).

 figure: Fig. 3.

Fig. 3. (a) the fringe pattern; (b) the unwrapped phase with segment piston error; (c) segment piston error.

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The initial idea is to solve the problem by minimizing the Zernike polynomials fitting residuals (MZR), because the Zernike polynomial is a good representation of a wavefront and the segment piston is different from the aberration representation of the Zernike polynomial terms. However, the selection of the number of Zernike polynomial terms in this method has a great impact on the correctness of the results. Underfitting and overfitting are easy to occur, resulting in low accuracy of the results, especially for the phase with complex aberration forms. So we started looking for other methods.

We notice that the difference between the upper right corner segment and its neighbors will be larger than the difference between the other segments, as shown in Fig. 3(b). The gradient of phase is used to express this difference. Combined with the idea of Zernike fitting, we come up with the idea of using minimization of Zernike gradient polynomial residual (MZGR) to calculate segment piston errors.

2.2 Description of the method based on minimization of Zernike gradient polynomial residual

In this paper, we use Noll Zernike polynomials. The polynomials can be written as [2]

$${Z_j}(\rho ,\theta ) = \sqrt {\frac{{2({n + 1} )}}{{1 + {\delta _{m0}}}}} R_n^m(\rho )\left\{ {\begin{array}{c} {\cos }\\ {\sin } \end{array}} \right\}({m\theta } ),$$
where j is the general index of Zernike polynomials. n is the power of the radial coordinate. m is the multiplication factor of the angular coordinate. And δm0 is the Kronecker delta.

Because segment pistons are different among different phase segments, we need to label segments, like Fig. 4. The segment piston Pk(x, y) of the phase segment k can be defined as

$${P_k}(x,y) = \left\{ {\begin{array}{c} {1,(x,y) \in Part(k),}\\ {0,(x,y) \in others,\textrm{ }} \end{array}} \right.$$

 figure: Fig. 4.

Fig. 4. Labels k of segments cut by spiders.

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The phase with segment piston error φ(x, y) and the real phase φ0(x, y) can be given by

$$\begin{array}{c} \varphi (x,y) = \sum\limits_N {{A_j}{Z_j}(x,y)} + \sum\limits_M {{B_k}{P_k}(x,y)} + \varepsilon (x,y),\\ {\varphi _0}(x,y) = \sum\limits_N {{A_j}{Z_j}(x,y)} + \varepsilon (x,y), \end{array}$$
where Aj and Zj are the j-th terms of Zernike coefficient and polynomial, and ε(x, y) is the residual of the phase after Zernike polynomials fitting. Bk is the coefficient of the segment piston Pk. N is the maximum number of Zernike polynomials. M is the number of phase segments.

The above mentioned that the gradient of the phase will have no segment piston. The gradient of φ(x, y) can be written as Eq. (4), and the gradient of φ0 (x, y) is the same as it.

$$\boldsymbol{\nabla }\varphi (x,y) = \sum\limits_N {{A_j}\boldsymbol{\nabla }{Z_j}(x,y)} + \boldsymbol{\nabla }\varepsilon (x,y).$$

Chunyu Zhao and James H. Burge [20] proposed a way to acquire orthonormal vector polynomials S as functions of Zernike gradients. And S can be given by

$$\begin{array}{cc} {{\textbf{S}_j} = \frac{1}{{\sqrt {2n(n + 1)} }}\boldsymbol{\nabla }{Z_j}}&{n = m}\\ {{\textbf{S}_j} = \frac{1}{{\sqrt {4n(n + 1)} }}\left( {\boldsymbol{\nabla }{Z_j} - \sqrt {\frac{{n + 1}}{{n - 1}}} \boldsymbol{\nabla }{Z_{j^{\prime}(n - 2,m)}}} \right)}&{n \ne m} \end{array}$$
where j’ is the term of whom the power of the radial coordinate is n-2 and the multiplication factor of the angular coordinate is m, and j-j’ is even when m ≠ 0. We can use these polynomials to rewrite Eq. (4),
$$\boldsymbol{\nabla }\varphi (x,y) = \sum\limits_N {{\alpha _i}{\textbf{S}_i}(x,y)} + \boldsymbol{\nabla }\varepsilon (x,y).$$

The solution of Eq. (6) can be regarded as a least squares problem, as shown in Eq. (7).

$$\min {,\|{\boldsymbol{\nabla }\varepsilon (x,y)} \|^2} = \min {\left\|{\boldsymbol{\nabla }\varphi (x,y) - \sum\limits_N {{\alpha_i}{\textbf{S}_i}(x,y)} } \right\|^2}$$

S and ∇φ are all vectors, which are complex for calculation. To simplify the calculation, Eq. (6) can be split into two pieces,

$$\begin{aligned} &\frac{{d\varphi }}{{dx}}(x,y) = \sum\limits_N {{\alpha _i}{S_{ix}}(x,y)} + {\varepsilon _x}(x,y);\\ &\frac{{d\varphi }}{{dy}}(x,y) = \sum\limits_N {{\alpha _i}{S_{iy}}(x,y)} + {\varepsilon _y}(x,y). \end{aligned}$$

Rewrite Eq. (8) in a matrix case,

$$\begin{aligned} &{\left[ {\begin{array}{c} {{S_{1x(1,1)}}, \cdots ,{S_{Nx(1,1)}}}\\ \vdots \\ {{S_{1x(p,q)}}, \cdots ,{S_{Nx(p,q)}}} \end{array}} \right] \cdot [{\alpha _2}, \cdots ,{\alpha _N}]^{\prime} = \left[ {\begin{array}{c} {\frac{{d\varphi }}{{dx}}(1,1)}\\ \vdots \\ {\frac{{d\varphi }}{{dx}}(p,q)} \end{array}} \right] - \left[ {\begin{array}{c} {{\varepsilon_x}(1,1)}\\ \vdots \\ {{\varepsilon_x}(p,q)} \end{array}} \right]}\\ &{\left[ {\begin{array}{c} {{S_{1y(1,1)}}, \cdots ,{S_{Ny(1,1)}}}\\ \vdots \\ {{S_{1y(p,q)}}, \cdots ,{S_{Ny(p,q)}}} \end{array}} \right] \cdot [{\alpha _2}, \cdots ,{\alpha _N}]^{\prime} = \left[ {\begin{array}{c} {\frac{{d\varphi }}{{dy}}(1,1)}\\ \vdots \\ {\frac{{d\varphi }}{{dy}}(p,q)} \end{array}} \right] - \left[ {\begin{array}{c} {{\varepsilon_y}(1,1)}\\ \vdots \\ {{\varepsilon_y}(p,q)} \end{array}} \right]} \end{aligned},$$
where p, q are the number of rows and columns of the wavefront matrix, respectively. Equation (9) is a typical least square method problem, so we can calculate αj easily. In addition, the coefficients Aj of Zernike polynomials and the coefficients αj of S polynomials have the following relation [20]
$$\begin{aligned} &{{A_j} = \frac{{{\alpha _{j(n,m)}}}}{{\sqrt {2n(n + 1)} }} - \frac{{{\alpha _{j^{\prime}(n + 2,m)}}}}{{\sqrt {4(n + 1)(n + 9)} }}}&{n = m}\\ &{{A_j} = \frac{{{\alpha _{j(n,m)}}}}{{\sqrt {4n(n + 1)} }} - \frac{{{\alpha _{j^{\prime}(n + 2,m)}}}}{{\sqrt {4(n + 1)(n + 9)} }}}&{n \ne m} \end{aligned}$$

We can directly obtain the coefficients Aj of Zernike polynomials except for the first Zernike polynomial term from Eq. (10). And the coefficients of piston A1 and segment pistons Bk can be calculated by

$$\left[ \begin{array}{c} {Z_{1(1,1)}},{P_{1(1,1)}}, \cdots ,{P_{M(1,1)}}\\ \vdots \\ {Z_{1(p,q)}},{P_{1(p,q)}}, \cdots ,{P_{M(p,q)}} \end{array} \right] \cdot [{A_1},{B_1}, \cdots ,{B_M}]^{\prime} = \left[ {\begin{array}{c} {\varphi (1,1) - \sum\limits_N {{A_j}{Z_j}(1,1)} }\\ \vdots \\ {\varphi (p,q) - \sum\limits_N {{A_j}{Z_j}(p,q)} } \end{array}} \right] - \left[ {\begin{array}{c} {\varepsilon (1,1)}\\ \vdots \\ {\varepsilon (p,q)} \end{array}} \right],$$

Therefore, the recovered phase φr(x, y) can be written as,

$${\varphi _r} = \varphi (x,y) - \sum\limits_M {{B_k}{P_k}(x,y)} .$$

2.3 Parameter selection and evaluation method

During calculation, the choice of the number of Zernike gradient polynomials fitting terms, N, needs to be considered. If N is too small, the results may be underfitting and the segment piston may not be moved completely. If N is too lager, the calculation time will be prolonged and overfitting may occur. So, we need an evaluation method to verify whether N is suitable and whether the recovered phase is right. To simplify this process, the power of the radial coordinate n is considered instead of N.

For Eq. (7), the aim of MZGR is to minimize the residual. So ε(x, y) can be the criterion for the correctness of the results. Define variable R as Eq. (13),

$$R = \frac{{RMS(\varepsilon (x,y))}}{{RMS({\varphi _r}(x,y))}} = 1 - \frac{{RMS\left( {\sum\limits_{j = 1}^h {{A_j}{Z_j}(x,y)} + \sum\limits_{k = 1}^l {{B_k}{P_k}(x,y)} } \right)}}{{RMS({\varphi _r}(x,y))}},R \to 0.$$

The threshold of R should be considered. We can simply think that the segment piston is caused by the difference between the edge points of the two segments divided by the spider exceeding π. So, we can simply assume that in the area covered by spiders, the PV (peak to valley) of the phase is π. And it becomes π/2 due to the double-path of the testing optical path. Assuming that the spider is 1/10 the width of the aperture, the PV of the whole aperture should be 5π for the tilt aberration as an example. In that case, the RMS of the whole aperture is 0.647 × 2π. In general, RMS of the residual ε(x, y) is small during the system inspection phase, so we assume it is 0.1 × 2π. According to Eq. (13), R should be 0.15. Because this is a loose calculation, the threshold of R can be smaller. In this paper, we take R = 0.1. In addition, using the phase shown in Fig. 2, we can draw the relationship between R and n to estimate the threshold of R, shown in Fig. 5. We can see that R finally stabilizes below 0.1, which is consistent with the previous calculation. So, 0.1 can be a good choice for the threshold of R.

 figure: Fig. 5.

Fig. 5. The relationship between R and n of Fig. 2(a), (b).

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The flowchart of the algorithm based on Zernike gradient fitting is shown in the Fig. 6.

 figure: Fig. 6.

Fig. 6. Flowchart of the algorithm based on Zernike gradient fitting.

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3. Numerical simulation and analysis

We simulated the phase of an optical system with obscurations using Zernike polynomials to demonstrate the advantage and the validity of this method. Three common forms of obscurations are used to simulate, shown in Fig. 7, and the random first 37 terms of Zernike coefficients of the simulated phase (the coefficients range from −0.5 to 0.5) are shown in Table 1. The results are presented in Fig. 8. From Fig. 8(b)∼(d), the difference of them is hard to see directly, but the residuals are a good way to compare the difference. The residual is the difference between the results calculated by one algorithm and the theoretical phase. According to Fig. 8(f1)-(f3), it can be found that the discontinuous phases only unwrapped by DCT (discrete cosine transform) algorithm [10] have segment piston errors. MZR algorithm does not remove the segment piston error correctly, as shown in Fig. 8(g1)-(g3). And there is bare of residual and that residual comes from the unwrapping rather than the removing segment piston in Fig. 8(h1)-(h3), which demonstrates the availability of the MZGR method to remove segment piston.

 figure: Fig. 7.

Fig. 7. Three common forms of obscurations used to simulate.

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 figure: Fig. 8.

Fig. 8. Results of phase unwrapping and removing segment pistons. (a1)-(a3): theoretical true phase cut by different spiders; (b1)-(b3): the phase unwrapped by DCT algorithm; (c1)-(c3): the phase calculated by MZR; (d1)-(d3): the phase calculated by MZGR; (e1)-(e4): the fringe pattern; (f1)-(f3): the residuals of DCT; (g1)-(g3):the residuals of MZR; (h1)-(h3):the residuals of MZGR.

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Tables Icon

Table 1. The first 37 terms of Zernike coefficients with random values

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Moreover, in order to test the performance of the method under various aberrations, we simulated 100 phases with different random coefficients of the first 37 terms of Zernike polynomials for every obscuration shown in Fig. 7, and the simulation results all performed well. The residuals of all results are close to 0, as shown in Fig. 9. The results of their obscurations are also close to 0. For images of size 2048 × 2048, the algorithm takes 0.36 seconds to run.

 figure: Fig. 9.

Fig. 9. R of all the 100 simulation results: (a) the same obscuration as Fig. 7(a); (b) the same obscuration as Fig. 7(b); (c) the same obscuration as Fig. 7(c).

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Subsequently, we made the obscuration more complex and added Gaussian noise with a standard deviation of 0.08 rad. The result is illustrated in Fig. 10, which proves that the method can still calculate the segment piston error even though the obscuration is complex and the phase has noise.

 figure: Fig. 10.

Fig. 10. Results of phase unwrapping and removing segment pistons. (a): theoretical true phase cut by different spiders; (b): the phase unwrapped by DCT algorithm; (c): the phase calculated by MZR; (d3): the phase calculated by MZGR; (e): the fringe pattern; (f): the residuals of DCT; (g): the residuals of MZR; (h): the residuals of MZGR.

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4. Experiment

To verify the validity and practicality of this method, we made three common forms of obscurations and installed them in the same transmission system. The experiment layout and three obscurations are shown in Fig. 11 and Fig. 12.

 figure: Fig. 11.

Fig. 11. Experiment layout.

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 figure: Fig. 12.

Fig. 12. Sketch and facility of three obscurations. (a) cross obscuration; (b) trisection obscuration; (c) tangent obscuration.

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We test the system with and without obscuration for each obscuration. The result without the obscuration is the referenced phase since there is no segment piston. Each wavefront is removed the first 9 terms of Zernike polynomials to make segment piston more obvious. Figure 13–15 show the test results and recovered results of three obscurations. From Fig. 13–15(d) (e), we can see the recovered phases have no segment piston and the RMS of the residuals is decreased approximately one order of magnitude (0.154 λ to 0.020 λ, 0.142 λ to 0.017 λ, 0.131 λ to 0.027 λ, λ=632.8 nm), which can prove the effectiveness of the method.

 figure: Fig. 13.

Fig. 13. Cross obscuration test result and recovered result.

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 figure: Fig. 14.

Fig. 14. Trisection obscuration test result and recovered result.

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 figure: Fig. 15.

Fig. 15. Tangent obscuration test result and recovered result.

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5. Conclusions

When the phase is obstructed by spiders, segment piston errors can appear in the phase unwrapping result. In this paper, we presented a method based on minimization of Zernike gradient polynomial residual to calculate the segment piston errors. Our simulation and experiment results proved the effectiveness of MZGR and demonstrate that this method is applicable to the phase with a variety of common obstructions and complex aberrations.

Funding

National Natural Science Foundation of China (62127901, 61805243, 62075218, 12003034, 62005278); Bureau of International Cooperation, Chinese Academy of Sciences (181722KYSB20180015); Youth Innovation Promotion Association of the Chinese Academy of Sciences (2019221); Key Research Program of Frontier Science, Chinese Academy of Sciences (QYZDJ-SSW-JSC038).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but maybe obtained from the authors upon reasonable request.

References

1. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital Wavefront Measuring Interferometer for Testing Optical Surfaces and Lenses,” Appl. Opt. 13(11), 2693–2703 (1974). [CrossRef]  

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4. D. C. Ghiglia and M. D. Pritt, Two-dimensional phase unwrapping: theory, algorithms, and software (A Wiley Interscience Publication, 1998), Chap. 1.

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9. H. Takajo and T. Takahashi, “Least-squares phase estimation from the phase difference,” J. Opt. Soc. Am. A 5(3), 416–425 (1988). [CrossRef]  

10. D. Kerr, G. H. Kaufmann, and G. E. Galizzi, “Unwrapping of interferometric phase-fringe maps by the discrete cosine transform,” Appl. Opt. 35(5), 810–816 (1996). [CrossRef]  

11. Y. Guo, X. Chen, and T. Zhang, “Robust phase unwrapping algorithm based on least squares,” Opt. Lasers Eng. 63, 25–29 (2014). [CrossRef]  

12. C. Wei, J. Ma, X. Miao, N. Wang, Y. Zong, and C. Yuan, “Residue calibrated least-squares unwrapping algorithm for noisy and steep phase maps,” Opt. Express 30(2), 1686–1698 (2022). [CrossRef]  

13. C. Zuo, Q. Chen, L. Huang, and A. Asundi, “Phase discrepancy analysis and compensation for fast Fourier transform based solution of the transport of intensity equation,” Opt. Express 22(14), 17172–17186 (2014). [CrossRef]  

14. L. Huang, C. Zuo, M. Idir, W. Qu, and A. Asundi, “Phase retrieval with the transport-of-intensity equation in an arbitrarily shaped aperture by iterative discrete cosine transforms,” Opt. Lett. 40(9), 1976–1979 (2015). [CrossRef]  

15. N. Pandey, A. Ghosh, and K. Khare, “Two-dimensional phase unwrapping using the transport of intensity equation,” Appl. Opt. 55(9), 2418–2425 (2016). [CrossRef]  

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but maybe obtained from the authors upon reasonable request.

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Figures (15)
Fig. 1.
Fig. 1. (a) fringe pattern; (b) wrapped phase; (c) unwrapped phase.

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Fig. 2.
Fig. 2. Wavefront test results with segment pistons of commercial interferometers(removed the first nine Zernike polynomials) (a) 4D PhaseCam 6010; (b) ZYGO Verfire HDX.

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Fig. 3.
Fig. 3. (a) the fringe pattern; (b) the unwrapped phase with segment piston error; (c) segment piston error.

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Fig. 4.
Fig. 4. Labels k of segments cut by spiders.

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Fig. 5.
Fig. 5. The relationship between R and n of Fig. 2(a), (b).

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Fig. 6.
Fig. 6. Flowchart of the algorithm based on Zernike gradient fitting.

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Fig. 7.
Fig. 7. Three common forms of obscurations used to simulate.

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Fig. 8.
Fig. 8. Results of phase unwrapping and removing segment pistons. (a1)-(a3): theoretical true phase cut by different spiders; (b1)-(b3): the phase unwrapped by DCT algorithm; (c1)-(c3): the phase calculated by MZR; (d1)-(d3): the phase calculated by MZGR; (e1)-(e4): the fringe pattern; (f1)-(f3): the residuals of DCT; (g1)-(g3):the residuals of MZR; (h1)-(h3):the residuals of MZGR.

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Fig. 9.
Fig. 9. R of all the 100 simulation results: (a) the same obscuration as Fig. 7(a); (b) the same obscuration as Fig. 7(b); (c) the same obscuration as Fig. 7(c).

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Fig. 10.
Fig. 10. Results of phase unwrapping and removing segment pistons. (a): theoretical true phase cut by different spiders; (b): the phase unwrapped by DCT algorithm; (c): the phase calculated by MZR; (d3): the phase calculated by MZGR; (e): the fringe pattern; (f): the residuals of DCT; (g): the residuals of MZR; (h): the residuals of MZGR.

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Fig. 11.
Fig. 11. Experiment layout.

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Fig. 12.
Fig. 12. Sketch and facility of three obscurations. (a) cross obscuration; (b) trisection obscuration; (c) tangent obscuration.

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Fig. 13.
Fig. 13. Cross obscuration test result and recovered result.

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Fig. 14.
Fig. 14. Trisection obscuration test result and recovered result.

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Fig. 15.
Fig. 15. Tangent obscuration test result and recovered result.

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Tables (1)

Tables Icon

Table 1. The first 37 terms of Zernike coefficients with random values

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Equations (13)

Equations on this page are rendered with MathJax. Learn more.

Z j ( ρ , θ ) = 2 ( n + 1 ) 1 + δ m 0 R n m ( ρ ) { cos sin } ( m θ ) ,
P k ( x , y ) = { 1 , ( x , y ) ∈ P a r t ( k ) , 0 , ( x , y ) ∈ o t h e r s ,  
φ ( x , y ) = ∑ N A j Z j ( x , y ) + ∑ M B k P k ( x , y ) + ε ( x , y ) , φ 0 ( x , y ) = ∑ N A j Z j ( x , y ) + ε ( x , y ) ,
∇ φ ( x , y ) = ∑ N A j ∇ Z j ( x , y ) + ∇ ε ( x , y ) .
S j = 1 2 n ( n + 1 ) ∇ Z j n = m S j = 1 4 n ( n + 1 ) ( ∇ Z j − n + 1 n − 1 ∇ Z j ′ ( n − 2 , m ) ) n ≠ m
∇ φ ( x , y ) = ∑ N α i S i ( x , y ) + ∇ ε ( x , y ) .
min , ‖ ∇ ε ( x , y ) ‖ 2 = min ‖ ∇ φ ( x , y ) − ∑ N α i S i ( x , y ) ‖ 2
d φ d x ( x , y ) = ∑ N α i S i x ( x , y ) + ε x ( x , y ) ; d φ d y ( x , y ) = ∑ N α i S i y ( x , y ) + ε y ( x , y ) .
[ S 1 x ( 1 , 1 ) , ⋯ , S N x ( 1 , 1 ) ⋮ S 1 x ( p , q ) , ⋯ , S N x ( p , q ) ] ⋅ [ α 2 , ⋯ , α N ] ′ = [ d φ d x ( 1 , 1 ) ⋮ d φ d x ( p , q ) ] − [ ε x ( 1 , 1 ) ⋮ ε x ( p , q ) ] [ S 1 y ( 1 , 1 ) , ⋯ , S N y ( 1 , 1 ) ⋮ S 1 y ( p , q ) , ⋯ , S N y ( p , q ) ] ⋅ [ α 2 , ⋯ , α N ] ′ = [ d φ d y ( 1 , 1 ) ⋮ d φ d y ( p , q ) ] − [ ε y ( 1 , 1 ) ⋮ ε y ( p , q ) ] ,
A j = α j ( n , m ) 2 n ( n + 1 ) − α j ′ ( n + 2 , m ) 4 ( n + 1 ) ( n + 9 ) n = m A j = α j ( n , m ) 4 n ( n + 1 ) − α j ′ ( n + 2 , m ) 4 ( n + 1 ) ( n + 9 ) n ≠ m
[ Z 1 ( 1 , 1 ) , P 1 ( 1 , 1 ) , ⋯ , P M ( 1 , 1 ) ⋮ Z 1 ( p , q ) , P 1 ( p , q ) , ⋯ , P M ( p , q ) ] ⋅ [ A 1 , B 1 , ⋯ , B M ] ′ = [ φ ( 1 , 1 ) − ∑ N A j Z j ( 1 , 1 ) ⋮ φ ( p , q ) − ∑ N A j Z j ( p , q ) ] − [ ε ( 1 , 1 ) ⋮ ε ( p , q ) ] ,
φ r = φ ( x , y ) − ∑ M B k P k ( x , y ) .
R = R M S ( ε ( x , y ) ) R M S ( φ r ( x , y ) ) = 1 − R M S ( ∑ j = 1 h A j Z j ( x , y ) + ∑ k = 1 l B k P k ( x , y ) ) R M S ( φ r ( x , y ) ) , R → 0.
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